Optimal line of play in a false carding situation

This is a slightly revised version of an article I posted in rec.games.bridge
in 1987.

The situation:

The board has xxx of trump; there is no problem with entries to the board.
On the first round the declarer leads from the board and plays the A from
his hand. The LHO has 9x or Tx in her hand. According to the books (as
they were in 1987) LHO has must make an obligatory false card. The
reasoning is that if the declarer has five trump headed by the AKJ (the
only relevant case) then he will take a winning finesse on the second
round of trump unless he can be bamboozled into playing for the drop.

The book play is not best however.
The relevant cases
that I considered are (a) declarer has AKJTx, (b) AKJ9x, and (c) AKJxx.
(The three cases must be considered in conjunction.) If the declarers
objective is to bring the suit in without loss, best play on both sides
runs as follows:

On the first round:

On the second round:

(a) If LHO has played small, RHO should false card with probability q >= 1/6
from 9xx or Txx.

(b) If LHO has played 9 or T, RHO should false card from 9xx or Txx with
probability r, 1/2 <= r <= 2p.

Declarers play:

If declarer has AKJTx or AKJxx declarer finesses on second round.

If declarer has AKJ9x then

(a) Declarer finesses the 9 against

Round 1: xxAx
Round 2: xx

(b) Declarer finesses the J against

Round 1: xxAx
Round 2: xT

(c) Declarer plays for the drop against

Round 1: xxAT
Round 2: xx

Declarer's play is minimax -- opponents cannot improve by altering their
falsecarding. However declarer can gain against erroneous false carding
as follows:

I. Declarer has AKJTxx

If LHO false cards from 9x with p<1/18, declarer should play for
the drop if LHO plays the 9 on the first round.

II. Declarer has AKJ9x

(a) If RHO false cards with probability q<1/6 from Txx then the declarer
should play for the drop on

Round 1: xxAx
Round 2: xx

(b) If LHO false cards with probability p<1/6 from Tx then the declarer
should finesse the J on

Round 1: xxAx
Round 2: xx

(c) If LHO false cards with probability p>1/3 from Tx then the declarer
should finesse the J on

Round 1: xxAT
Round 2: xx

III. Declarer has AKJxx

(a) If LHO false cards with probability p>1/2 from 9x or Tx the declarer
should play for the drop against

Round 1: xxAx
Round 2: x9/T

(b) If RHO false cards with probability r<1/2 from 9xx or Txx then the
declarer should play for drop against

Round 1: xxA9/T
Round 2: xx

(c) If r>2p then the declarer should play for the drop against

Round 1: xxA9 or Round 1: xxAT
Round 2: xT Round 2: x9

Since the books say that the false card from Tx is obligatory
and do not mention the false card by RHO it is probably more profitable
to assume that cases II.a, II.c, III.a, and III.b apply in duplicate or
tournament play. Here are the relevant percentages:

Hand:

A

B

C

AKJ9x

1144/2300

1209/2300

1027/2300

AKJxx

845/2300

923/2300

793/2300

Column A is the probability of bringing in the suit without loss if
both sides play correctly. Column B is the probability if the declarer
assumes that opponents are following 'book'. Column C is the probability
if declarer assumes that the opponents are following 'book' and they
are, in fact, playing correctly.

Note: I have assumed in the analysis that the location of the 8 does not
matter and that the 9 and the T may be treated as equals.